New Perturbation-Iteration Solutions For Bratu-Type Equations
Yiğit Aksoy, Mehmet Pakdemirli and Hakan Boyacı.

Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye Manisa, Turkey. (Tel: +90(236)2412146
(office), e-mail: yigit.aksoy@ bayar.edu.tr, [email protected] [email protected] ).
Abstract: Perturbation-iteration theory is systematically generated for both linear and nonlinear second
order differential equations and applied to Bratu-type equations. Different perturbation-iteration
algorithms depending upon the number of Taylor expansion terms are proposed. Using the iteration
formulas derived by different perturbation-iteration algorithms, new solutions of Bratu type equations are
obtained. Solutions constructed by different perturbation-iteration algorithms are contrasted with each
other as well as with numerical solutions. It is found that algorithms with more Taylor series expansion
terms yield more accurate results.
Keywords: Perturbation Methods, Perturbation-Iteration Algorithms, Bratu’s equations.

1. INTRODUCTION
Perturbation Methods (Nayfeh 1981) are one of the most
common approximate methods in study of nonlinear
mathematical models arising in physics and engineering. A
major limitation of the method is the small parameter
restriction which makes the solutions valid for weakly
nonlinear systems. To overcome this limitation, iterationperturbation methods are proposed in the literature to validate
solutions for strongly nonlinear problems (He 2006) in where
the nonlinear terms are linearized by substitution of iterative
solution functions from previous iteration results. Usually, in
iteration-perturbation methods, the equations are cast into an
alternative form before applying the iteration procedure.
Some of the algorithms developed can only work for specific
problems. A general approach valid for all types of equations
which do not require non-standard pre-transformations and
initial assumptions is lacking in the literature.
The aim in this study is to develop new perturbation-iteration
algorithms applicable to a wide range of equations which do
not require special transformations and initial assumptions.
Motivated by the results of perturbation-iteration algorithms
for algebraic equations (Pakdemirli and Boyacı). The basic
logic is extended to second order differential equations in this
study. Although iteration-perturbation and perturbationiteration algorithms both contain perturbations and iterations
over the perturbative solutions, the perturbation-iteration
algorithms developed in this work are new and different from
the existing literature on iteration-perturbation methods.
To illustrate the accuracy of proposed perturbation-iteration
method, well known Bratu’s boundary and initial value
problems are selected as examples.
2. PERTURBATION-ITERATION ALGORITHM PIA (1,1)
In this section, a perturbation-iteration algorithm is developed
by taking one correction term in the perturbation expansion
and correction terms of only first derivatives in the Taylor
Series expansion, i.e. n=1, m=1. The algorithm is named PIA
(1,1). Consider a second order differential equation,
F(u, u, ε)  0
(1)
with u=u(t) and ε the perturbation parameter. Only one
correction term is taken in the perturbation expansion.
u n 1  u n  ε (u c )n
(2)
Upon substitution of (2) into (1) and expanding in a Taylor
series with first derivatives only yields
F(un , u n ,0)  Fu (un , u n ,0) ε (u c ) n
 Fu  (un , u n ,0) ε (uc ) n  Fε (un , u n ,0) ε  0
where
(3)
( ) denote differentiation with respect to the
independent variable and Fu 
F
F
F
, Fu  
, Fε 
.
u 
u
ε
Reorganizing the equation
(uc ) n 
Fε 
Fu
(u c ) n  
Fu 
Fu 
F
ε
(4)
and keeping in mind that all derivatives are evaluated at
ε =0, it is readily observed that the above equation is a
variable coefficient linear second order differential equation.
Starting with an initial guess u0 , first (uc)0 is calculated from
(4) and then substituted into (2) to calculate u1. The iteration
procedure is repeated using (4) and (2) until a satisfactory
result is obtained.
3. PERTURBATION-ITERATION ALGORITHM PIA (1,2)
In this section, a perturbation-iteration algorithm is obtained
by taking one correction term in the perturbation expansion
and correction terms up to second derivatives in the Taylor
Series expansion, i.e. n=1, m=2. The algorithm is called
PIA(1,2). The expansion with one correction term is
u n 1  u n  ε (u c )n
(5)
(uc ) n  2u n  (un  2)
For the initial assumed function, one may take a trivial
solution which satisfies the given initial conditions
u0  0
(12)
and using Equation (11) and (2) , the approximate solutions
at each step are
which upon substitution into (1) and expanding in a Taylor
series up to second order derivatives yields after arrangement
u1  x 2
F(un , u n ,0)  Fu  (un , u n ,0) ε (uc ) n
 Fu (un , u n ,0) ε (u c ) n  Fε (un , u n ,0) ε
u2  x2 
1
1
 Fεε (un , u n ,0)ε 2  Fu u  (un , u n ,0) ε 2 (uc ) 2n
2
2
1
 Fuu (un , u n ,0)ε 2 (u c ) 2n
2
 Fu u (un , u n ,0)ε 2 (uc ) n (u c ) n
u3  x 2 
(6)
(14)
x4 x6

6 90
(15)
x4 x6
x8
x10
u5  x   

6 90 2520 113400
 Fuε (un , u n ,0)ε (u c ) n  0
(16)
6
Starting from x coefficient, the coefficients of higher order
terms deviate from the Taylor expansion of the exact
solution.
Reorganizing the equation yields
(7)
4.1.2. Perturbation Iteration Algorithm PIA (1,2)
For this case, Equation (10) will be treated by perturbationiteration algorithm PIA (1,2). The terms in (7) are evaluated
and then selecting ε =1, the equation takes the simplified
form
4. BRATU-TYPE PROBLEMS
The new algorithms developed will be applied to Bratu type
differential equations in this section. Initial and Boundary
value problem will be treated.
(u c ) n  2(u c ) n  2u n  u 2n  (u n  2)
Consider Bratu’s initial value problem
(17)
For initial assumption
u0  0
4.1. Example Problem 1
(18)
Substituting this function to (17) yields
u
(8)
(u c ) 0  2(u c ) 0  2
(19)
Solving (19), substituting into (5) and applying the initial
conditions yields
for which the exact solution is
u  2ln(cosx)
x4
6
Progressing in a similar way
2
u  2e  0, 0  x  1
u(0)  u(0)  0
(13)
2
 Fu ε (un , u n ,0)ε 2 (uc ) n
(ε F  ε 2 F )(u c ) n  (ε Fu  ε 2 Fuε )(uc ) n
u 
u ε
1
2
2
 F
ε (u c ) 2
n  Fu u ε (u c ) n (u c ) n
2 u u 
1
2
 1 Fuu ε 2 (u c ) 2
n  F  Fε ε  2 Fεε ε
2
(11)
(9)
4.1.1. Perturbation Iteration Algorithm PIA (1,1)
u1  Cosh( 2 x)  1
(20)
Using this function, the result for second iteration is
Rewrite Eq. (8) in the following form
F(u, u, ε)  u  2eε u  0
(10)
ε is an artificially introduced small parameter. Terms
Fu  0, Fε  2u n and
in iteration formula (4) are
setting ε =1 , equation (4) reduces to
where
1
[9  8Cosh( 2 x)
12
(21)
 Cosh(2 2 x)  6 2 x Sinh( 2 x) ]
u 2  Cosh( 2 x)  1 
Instead of a polynomial expansion as presented in PIA (1,1)
one retrieves a functional expansion. Calculations are more
involved in this algorithm and require the need of software
such as Mathematica.
4.1.3. Perturbation Iteration Algorithm PIA (1,3)
In the light of previous analysis, in this section, one may
develop a perturbation-iteration algorithm by taking one
correction term in the perturbation expansion and three
derivatives in the Taylor series , i.e. n=1, m=3. Only the
specific form of PIA (1,3) for this problem is presented here
for brevity
u3
(uc )n  2(1  u n )(u c ) n  2u n  u 2n  n
3
 (un  2)
(22)
The initial trial function is selected as
u0  0
Bratu’s Initial Value Problem
Numerical
PIA (1,1)
x
Solution
u1
u2
u3
u1
u2
u1
u2
0.1
0.01001
0.01000
0.01001
0.01001
0.01001
0.01001
0.01001
0.01001
0.2
0.04026
0.04000
0.04026
0.04026
0.04026
0.04026
0.04026
0.04026
0.3
0.09138
0.09000
0.09135
0.09135
0.09135
0.09138
0.09135
0.09138
0.4
0.16445
0.16000
0.16426
0.16431
0.16431
0.16445
0.16431
0.16445
0.5
0.26116
0.25000
0.26041
0.26059
0.26059
0.26114
0.26059
0.26116
0.6
0.38393
0.36000
0.38160
0.38211
0.38212
0.38380
0.38212
0.38391
0.7
0.53617
0.49000
0.53001
0.53132
0.53134
0.53570
0.53134
0.53610
0.8
0.72278
0.64000
0.70826
0.71117
0.71124
0.72126
0.71124
0.72249
0.9
0.95088
0.81000
0.91935
0.92525
0.92542
0.94648
0.92542
0.94983
1
1.23125
1.00000
1.16667
1.17778
1.17818
1.21942
1.17818
1.22772
PIA (1,2)
PIA (1,3)
(23)
Table 1. Comparison of Perturbation-Iteration Algoritmsand
Numerical Results for Bratu’s Initial Value Problem.
(24)
4.2. Example Problem 2
The first iteration solution is
u1  Cosh( 2 x )  1
Consider Bratu’s first boundary value problem
Substituting this function to (22) yields
(uc )1  2(1  u1 )(u c )1  2u1  u12 
u13
3
u  λ e u  0 , 0  x  1
(25)
 (u1  2)
(27)
4.2.1. Perturbation Iteration Algorithm PIA (1,1)
Since the equation to be solved is a variable coefficient
equation which is involved, the function in the parentheses of
second term is approximated as 1 for simplicity. Solving
(25), substituting into the iteration expansion and applying
the boundary condition yields
1
[64
192
 63Cosh( 2 x)  Cosh(3 2 x)
u(0)  u(1)  0
u 2  Cosh( 2 x)  1 
Rewrite Eq. (27) in the following form
F(u, u, ε)  u  λ eε u  0
where is ε an artificially introduced small parameter.
Equation (4) reduces to
(uc ) n  λ u n  (un  λ)
(26)
 36 2 x Sinh( 2 x) ]
by calculating the relevant terms and setting
initial trial function is selected as
4.1.4 Comparisons with Numerical Solutions
u0  0
Comparisons of the different perturbation-iteration
algorithms with numerical solutions of equation (8) are given
in Table 1. All iteration algorithms rapidly converge to the
numerical solutions. Table 1 shows that PIA (1,3) performs
better than the others. Performance of PIA (1,2) is close to
PIA (1,3) and can be used also. Results deviate more as x
approaches 1. The percentage difference with numerical and
PIA (1,3) is appreciably small being only %0.29 however.
Note that three iteration terms are taken in PIA (1,1) whereas
only two iteration terms are taken in the others. Three
iteration solutions of PIA (1,1) are not better than the two
iteration solutions of PIA (1,2) and PIA (1,3).
(28)
(29)
ε  1 . The
(30)
and using equation (29) and (2) , the approximate solutions at
each step are
λ
u 1   (x 2  x)
2
(31)
λ
λ2
u 2   ( x 2  x )  ( x 4  2x 3  x )
2
24
(32)
λ
λ2
u 3   ( x 2  x )  ( x 4  2x 3  x )
2
24
3
λ

( x 6  3x 5  5x 3  3x )
720
4.2.3 Comparisons with Numerical Solutions
(33)
4.2.2. Perturbation Iteration Algorithm PIA (1,2)
For this case, equation (28) will be treated by perturbationiteration algorithm PIA (1,2). Equation (7) reads
λ
(u c ) n  λ(u c ) n   λ u n  u 2n  (u n  λ)
2
(34)
Starting with a trivial solution
u0  0
(35)
the first iteration solution is
 λ

u1  1  Cos(  x)  Sin(  x) Tan 

2


(36)
Using u1 as an initial approximation, the second iteration
solution is
 λ

u 2  1  Cos( λ x)  Sin( λ x) Tan 

2



 λ 
 λ
3 λ 
1
 - 15Cos
  3Cos

Sec3 



 2 

24
2
2






In Table 2, PIA (1,1) and PIA (1,2) results are compared with
the numerical ones in the whole domain. Three iterations are
taken in PIA (1,1) whereas only two iterations are taken in
PIA (1,2). Two iteration results of PIA (1,2) are better than
the three iteration results of PIA (1,1). Three or four digit
precision is achieved with only two term iterations of PIA
(1,2).
Table 2. Comparison of Perturbation-Iteration Algoritmsand
Numerical Results for Bratu’s Boundary Value Problem.
( λ  1)
Bratu’s First Boundary Value Problem
x
Numerical
Solutions
PIA (1,1)
PIA (1,2)
u1
u2
u3
u1
u2
0.1
0.04984
0.04500
0.04908
0.04949
0.04954
0.04983
0.2
0.08918
0.08000
0.08773
0.08851
0.08860
0.08915
0.3
0.11760
0.10500
0.11558
0.11665
0.11678
0.11756
0.4
0.13479
0.12000
0.13240
0.13365
0.13380
0.13473
0.5
0.14053
0.12500
0.13802
0.13934
0.13949
0.14048
0.6
0.13479
0.12000
0.13240
0.13365
0.13380
0.13473
0.7
0.11760
0.10500
0.11558
0.11665
0.11678
0.11756
0.8
0.08918
0.08000
0.08773
0.08851
0.08860
0.08915
0.9
0.04984
0.04500
0.04908
0.04949
0.04954
0.04983
5. CONCLUSIONS
The following conclusions can be derived from the study
 3
 3
 
 
 Cos   2x  λ   2Cos   x  λ 
 
 
 2
 2

 1
λ
 
 2Cos   x  λ   Cos (1  4 x)

2 
 
 2

1.
A systematic algorithmic approach to develop new
perturbation iteration algorithms is presented.
2.
The perturbation iteration algorithms developed do
not require “small perturbation parameter”
assumption as a prerequisite for valid solutions.
3.
 λ

 3x λ Cos( λ x) Sin 

 2 
The perturbation iteration algorithms are applied
successfully to Bratu type nonlinear problems and
iteration solutions with few steps converge to those
of numerical ones.
4.
 3
 
 3 x λ Sin   x  λ 
 
 2
PIA (1,2) and PIA (1,3) iteration solutions are of
functional type compared to polynomial type
solutions of PIA (1,1).
5.
Algebra involved in constructing iteration solutions
is more involved for PIA (1,2) and PIA (1,3)
compared to PIA (1,1).
6.
On the other hand, faster convergence is achieved
with PIA (1,2) and PIA (1,3) compared to PIA (1,1).

λ
 12 Cos (1  2 x)

2 


 λ
 Sin( λ x) 
 3(-2  3x) λ Cos 


 2 

(37)
7.
With the systematic approach given in this study,
new algorithms with PIA (n,m) ( n: number of
correction terms in the perturbation expansion; m:
order of derivatives in the Taylor series expansions
n  m ) can be constructed easily.
Acknowledgement- This work is supported by The
Scientific and Technological Research Council of Turkey
(TUBITAK) under project number 108M490.
REFERENCES
He, J. H. (2006). Some asymptotic methods for strongly
nonlinear equations, International Journal of Modern Physics
B, 20, 1141-1199.
Nayfeh, A.H. (1981) Introduction to
Techniques, John Wiley and Sons, New York.
Perturbation
Pakdemirli , M. and H. Boyacı (2007). Generation of Root
Finding Algorithms via Perturbation Theory and Some
Formulas, Applied Mathematics and Computation, 184, 783788.